Fluctuations of the vortex line density in turbulent flows of quantum fluids

Abstract

We present an analytical study of fluctuations of the vortex line density (VLD) $\ensuremath{\langle}\ensuremath{\delta}\mathcal{L}(\ensuremath{\omega})\ensuremath{\delta}\mathcal{L}(\ensuremath{-}\ensuremath{\omega})\ensuremath{\rangle}$ in turbulent flows of quantum fluids. Two cases are considered. The first is the counterflowing (Vinen) turbulence, where the vortex lines are disordered, and the evolution of quantity $\mathcal{L}(t)$ obeys the Vinen equation. The second case is the fluctuations of the VLD in a single vortex bundle, which develops inside the domain of the concentrated normal-fluid vorticity. The dynamics of the vortex bundle is described by the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations. The latter case is of special interest, because the set of the quantum vortex bundles is believed to mimic classical hydrodynamic turbulence. In steady states the VLD is related to the normal velocity as $\mathcal{L}={(\ensuremath{\rho}\ensuremath{\gamma}/{\ensuremath{\rho}}_{s})}^{2}{v}_{n}^{2}$ for the Vinen case. In the vortex bundle case, which appears inside the domain of a concentrated vorticity of normal fluid, the stationary quantity $\mathcal{L}$ can be found from the matching of velocities and is described by $\mathcal{L}=|\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}{\mathbf{v}}_{n}|/\ensuremath{\kappa}$. In nonstationary situations, and particularly in the fluctuating turbulent flow, there is a retardation between the instantaneous value of the normal velocity and the quantity $\mathcal{L}$. This retardation tends to decrease in accordance with the inner dynamics, which has a relaxation character. In both cases, the relaxation dynamics of the VLD is related to fluctuations of the relative velocity. However, for the Vinen case the rate of temporal change for $\mathcal{L}(t)$ is directly dependent upon $\ensuremath{\delta}{\mathbf{v}}_{ns}$, whereas for HVBK dynamics it depends on $\ensuremath{\nabla}\ifmmode\times\else\texttimes\fi{}\ensuremath{\delta}{\mathbf{v}}_{ns}$. Therefore, for the disordered case the spectrum $\ensuremath{\langle}\ensuremath{\delta}\mathcal{L}(\ensuremath{\omega})\ensuremath{\delta}\mathcal{L}(\ensuremath{-}\ensuremath{\omega})\ensuremath{\rangle}$ coincides with the spectrum ${\ensuremath{\omega}}^{\ensuremath{-}5/3}$. In the case of the bundle arrangement, the spectrum of the VLD varies (at different temperatures) from ${\ensuremath{\omega}}^{1/3}$ to ${\ensuremath{\omega}}^{\ensuremath{-}5/3}$ dependencies. This conclusion may serve as a basis for the experimental determination of what kind of turbulence is implemented in different types of generation.